Distortion

Fuji is relying on an image auto-correction strategy just like Micro-Four-Thirds - and this time they exploited it to the max. From a standard photographer perspective, using JPEGs or corrected RAWs, you don't need to worry - at 16mm there's a bit of barrel distortion of just under 1% and the issue is irrelevant from 23mm onward.
However, the situation changes dramatically when looking into uncorrected RAWs. At 16mm, the lens produces a barrel distortion of more than 7% (!!!) which is heading towards fisheye territory. Thus we can only conclude that the XC 16-50mm is heavily under-designed in this respect. At 23mm the issue has eased to ~2.3% and it's negligible beyond.
If you move your mouse cursor over the image you can switch to the corresponding "raw" results.

Vignetting

The image auto-correction has also tamed the vignetting characteristic. The light falloff remains below 0.5EV (f-stops) which is hardly noticeable. Funnily, the technical vignetting is higher at 16mm f/3.5 vs f/5.6 which would be an impossible behavior if it was a natural vignetting curve.

As to be expected the native characteristic is very different. The light falloff peaks at 1.6EV at 16mm f/3.5 and ~1EV at 50mm. Thus if needed (using uncorrected RAWs), you should stop down to f/8 at the extreme ends and f/5.6 in between.

MTF (resolution)

The Fujinon produced decent (native) resolution figures in the MTF lab. The center performance is generally very good to excellent from large apertures up to medium settings. At f/11 diffraction takes a somewhat higher toll thus reducing the quality ratings here. The border quality is mostly on a good to very good level. Same goes for the extreme corners in the lower focal length range. At 35mm and 50mm they are getting somewhat softer though.

The centering quality of the tested sample was Okay.

Below is a simplified summary of the formal findings. The chart shows line widths
per picture height (LW/PH) which can be taken as a measure for sharpness.
If you want to know more about the MTF50 figures you may check out the corresponding
Imatest Explanations

Chromatic Aberrations (CAs)

Lateral CA (color shadows at the image borders) are fairly pronounced albeit usually not excessive. They are rather heavy at 16mm @ f/3.5 with a peak beyond 2px on the average at the image borders. However, they are getting less relevant the more you stop down. Please note that the extreme corners show somewhat higher CAs than the measured borders though. At the other settings, the issue isn't quite as obvious with ~1.4px at 23mm @ f/4.2 and just over 1px at 50mm f/5.6. The 35mm setting isn't overly affected at all.